Differential Fault Analysis of the Advanced Encryption Standard Using a Single Fault

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Differential Fault Analysis of the Advanced Encryption Standard Using a Single Fault

Michael Tunstall, Debdeep Mukhopadhyay, Subidh Ali

To cite this version:

Michael Tunstall, Debdeep Mukhopadhyay, Subidh Ali. Differential Fault Analysis of the Advanced Encryption Standard Using a Single Fault. 5th Workshop on Information Security Theory and Prac- tices (WISTP), Jun 2011, Heraklion, Crete, Greece. pp.224-233, �10.1007/978-3-642-21040-2_15�.

�hal-01573310�

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Differential Fault Analysis of the Advanced Encryption Standard using a Single Fault

Michael Tunstall¹, Debdeep Mukhopadhyay², and Subidh Ali²

1 Department of Computer Science, University of Bristol, Merchant Venturers Building, Woodland Road,

Bristol BS8 1UB, United Kingdom.

[email protected]

2 Computer Sc. and Engg, IIT Kharagpur, India.

{debdeep,subidh}@cse.iitkgp.ernet.in

Abstract. In this paper we present a differential fault attack that can be applied to the AES using a single fault. We demonstrate that when a single random byte fault is induced at the input of the eighth round, the AES key can be deduced using a two stage algorithm. The first step has a statistical expectation of reducing the possible key hypotheses to 2³², and the second step to a mere 2⁸.

Keywords: Differential Fault Analysis, Fault Attack, Advanced Encryption Standard.

1 Introduction

The Advanced Encryption Standard (AES) [10] has been a de-facto standard for symmetric key cryptography since October 2000. Smart cards and secure microprocessors, therefore, typically include implementations of AES to protect the confidentiality and the integrity of sensitive information. To satisfy the high throughput requirements of such applications, these implementations are typically VLSI devices (crypto-accelerators) or highly optimized software routines (crypto-libraries).

Several applications of DFA to AES have been reported in the literature.

In [3], authors describe an analysis based on faults induced in one byte of the ninth round of AES that requires 250 faulty ciphertexts. An attack reported in [1] allows an attacker to recover the secret key with around 128 to 256 faulty ciphertexts. In [2], Dusart et al. show that using a fault which affects one byte anywhere between the eighth round MixColumn and ninth round MixColumn, an attacker would be able to derive the secret key using 40 faulty ciphertexts. The authors of [12] describe an attack on AES with single byte faults that requires two faulty outputs, where a fault is induced in the input of the eighth or ninth round, extended to one 32-bit fault in the ninth round in [8].

We can note that when the assumptions are on the value of a byte (either it being faulty or uncorrupted) the number of faulty pairs is quite small. However,

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it is difficult to be able to affect a given value with any certainty. When numerous faulty ciphertexts are required this problem is amplified, since an attacker needs to find a method of determining which faulty ciphertexts correspond to the desired model. We can, therefore, state that the attacks that are most likely to be realizable require the least faulty ciphertexts and assumptions on the effect of the fault.

In [9] a fault attack against AES was proposed, which suggested that a secret key can be derived using a singlebytefault induction at the input of the eighth round. The attack exploited the inter-relations between the fault values in the state matrix after the ninth roundMixColumnoperation and reduced the number of possible keys to around 2³². However it may be noted that this work, like the previous fault attacks on AES does not use the effect of the fault maximally in an information theoretic sense [7]. The work proposed in this paper improves the previous fault analysis on AES-128 and reduces the key space to its minimal possible set of hypotheses attainable using a single byte fault. In this paper, we describe the extended version of this attack, where an attacker could reduce the exhaustive search to 2⁸.

Notation

In this paper, multiplications are considered to be polynomial multiplications over F2⁸ modulo the irreducible polynomialx⁸+x⁴+x³+x+ 1. It should be clear from the context when a mathematical expression contains integer multi- plication.

Organization

The paper is organized as follows: In Section 2 we describe the background to this paper. In Section 3 we describe an attack based on one of the fault models given in Section 2. In Section 3 we extend this attack. In Section 4 we compare this paper to work described in the literature, and we conclude in Section 5.

2 Background

2.1 The Advanced Encryption Standard

The structure of the Advanced Encryption Standard (AES) , as used to per- form encryption, is illustrated in Algorithm 1. Note that we restrict ourselves to considering AES-128 and that the description above omits a permutation typically used to convert the plaintext P = (p1, p₂, . . . , p₁₆)(256) and key K = (k1, k₂, . . . , k₁₆)(256) into a 4×4 array of bytes, known as the state matrix. For example, the 128-bit plaintext input blockPwhich produces fault free (CT) and faulty ciphertexts (CT^′) are arranged in the following fashion

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Algorithm 1: The AES-128 encryption function.

Input: The 128-bit plaintext blockP and keyK.

Output: The 128-bit ciphertext blockC.

X←AddRoundKey(P, K) fori←1to10do

X←SubBytes(X) X←ShiftRows(X) if i6= 10then

X←MixColumns(X) end

K←KeySchedule(K) X←AddRoundKey(X, K) end

C←X returnC

P=







p1p5 p9 p13

p2p6 p10p14

p3p7 p11p15

p4p8 p12p16







CT=







x1 x5 x9 x13

x2 x6x10x14

x3 x7x11x15

x4 x8x12x16







CT^′=







x^′₁ x^′₅ x^′₉ x^′₁₃ x^′₂ x^′₆x^′₁₀x^′₁₄ x^′₃ x^′₇x^′₁₁x^′₁₅ x^′₄ x^′₈x^′₁₂x^′₁₆







wherexi∈ {0, . . . ,255} ∀i∈ {1, . . . ,16}. We also define the key matrix for the subkeys used in the ninth and tenth round asK₁₀={k1, . . . , k₁₆}andK₉={k₁^′, . . . , k^′₁₆}that are arranged in a state matrix as described above.

The encryption itself is conducted by the repeated use of a number of round functions:

– The SubBytes function is the only non-linear step of the block cipher. It is a bricklayer permutation consisting of an S-box applied to the bytes of the state.

Each byte of the state matrix is replaced by its multiplicative inverse, followed by an affine mapping. Thus the input bytexis related to the outputyof the S-Box by the relation, y = A x⁻¹ +B, where A and B are constant matrices. In the remainder of this paper we will refer to the functionS as the SubBytes function andS⁻¹ as the inverse of the SubBytes function.

– TheShiftRowsfunction is a byte-wise permutation of the state.

– TheKeySchedulefunction generates the next round key from the previous one.

The first round key is the input key with no changes, subsequent round keys are generated using the SubBytes function and XOR operations. This is shown in Algorithm 2 which shows how ther^th round key is computed from the (r−1)^th round key. The valuehr is a constant defined for ther^th round, and<<is used to denote a bitwise left shift.

– TheMixColumnis a bricklayer permutation operating on the state column by column. Each column of the state matrix is considered as a 4-dimensional vector where each element belongs toF(2⁸). A 4×4 matrixM whose elements are also in F(2⁸) is used to map this column into a new vector. This operation is applied on

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all the 4 columns of the state matrix. HereM and its inverseM⁻¹ are defined as:

M =





 2 3 1 1 1 2 3 1 1 1 2 3 3 1 1 2







M⁻¹=







14 11 13 9 9 14 11 13 13 9 14 11 11 13 9 14







All the elements inMandM⁻¹are elements ofF(2⁸) expressed as a decimal digit.

– AddRoundKey: Each byte of the array is XORed with a byte from a corresponding array of round subkeys.

Algorithm 2: The AES-128KeySchedulefunction.

Input: (r−1)^thround key (X =xifori∈ {1, . . . ,16}).

Output:r^th round keyX.

fori←0to3do

x(i<<2)+1←x(i<<2)+1⊕S(x(((i+1)∧3)<<2)+4) end

x1←x1⊕hr

fori←1to16do

if (i−1) mod 46= 0then xi←xi⊕xi−1

end end returnX

2.2 The Fault Model

The implementation of AES we target is an iterative one, i.e. where a round function is executed in a loop as described in Algorithm 1. An attacker can typically predict at what point in time certain events take place, e.g. when a particular round commences.

Moreover, the time certain events take can often be determined by analyzing a suitable side channel.

The fault model that we consider is the same as that used in many other papers, for example [9], where we assume that the effect of an induced fault is to change one byte to a random value.

For example, an attacker could attempt to use a glitch in the clock to create a fault at the input of a particular round with a certain probability. An iterative design helps in this regard, as the attacker is able to control the timing of fault induction by simply counting the number of clock edges from the start of an encryption.

3 The Fault Analysis

3.1 The First Step of the Fault Attack

If a fault is induced in a byte of the state matrix, which is then input to the eighth round, theMixColumnoperation at the end of the round propagates this fault to the

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entire column of the state. The ShiftRowoperation at the beginning of the following round will then shift these bytes to occupy different columns. The next MixColumn operation will then propagate the fault to the remaining twelve bytes.

This process is shown in Figure 1 where we show the diffusion of a byte fault induced at the input of the eighth round. The XOR difference of the state matrices of the two results, one fault free and the other faulty, is shown. This is what we use as basis for a differential fault analysis.

A1 A 2 A

3 A

4 A5 A

6 A

7 A

8 A9 A

10A 11A

12 A13A

14A 15A

16 A1 A

2 A

3 A 4 A6 A

7 A 8 A

5 A10 A9 A11A

12 A13A

14A A 15

16

F1

F 4

F 2 F 3 Round

Shift Row Eighth

F 3

F 3 F 1 2F1

F 1 3

F 4 F 4 F 1 3F 4 F4

2 F2

F 3 2

3 2F F 3 3 2F2

F2 F 1

F 4 F 3 F2 2f’

f’

3f’

Round Mix Column Eighth Round

Byte Sub f’

f

Eighth

Ninth Round Byte Sub

Tenth Round Byte Sub

Tenth Round Shift Row Ninth Round Mix Column

Ninth Round Shift Row f’

Fig. 1: Propagation of Fault Induced in the input of eighth round of AES.

If, given a fault in the input to the eighth round, we consider the state of the differences after the ninth round shift row, we can obtain the following set of equations that include the values of the key bytesk1,k8,k11 andk14, thus giving an expression for 32 bits ofK10.

2δ₁=S⁻¹(x1⊕k₁)⊕S⁻¹(x^′1⊕k₁) δ₁=S⁻¹(x14⊕k₁₄)⊕S⁻¹(x^′₁₄⊕k₁₄) δ₁=S⁻¹(x11⊕k₁₁)⊕S⁻¹(x^′₁₁⊕k₁₁) 3δ1=S⁻¹(x8⊕k8)⊕S⁻¹(x^′₈⊕k8)

,

Whereδ1,k1,k8,k11andk14 are all unknown values∈ {0, . . . ,255}.

The above system of equations can be used to reduce the possibilities for these 32 bits of the key. An attacker would select a value forδ1 and determine which values of k1,k8,k11 and k14 satisfy the equations using four independent exhaustive searches.

Each equation will return 0, 2, or 4 hypotheses [11]. If any of the four equations cannot be satisfied, i.e. there is an impossible differential [6], then any hypotheses for that value ofδ₁ can be discarded.

As noted in [4, 8] one can apply the same technique to recover information on the remaining bytes of the last sub key. That is, information on the remaining key bytes can be derived by using the following sets of equations: In order to obtain information

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onk2,k5,k12 andk15an attacker can use

3δ₂=S⁻¹(x5⊕k₅)⊕S⁻¹(x^′₅⊕k₅) 2δ2=S⁻¹(x2⊕k2)⊕S⁻¹(x^′₂⊕k2)

δ2=S⁻¹(x15⊕k15)⊕S⁻¹(x^′₁₅⊕k15) δ2=S⁻¹(x12⊕k12)⊕S⁻¹(x^′₁₂⊕k12) .

In order to obtain information onk3,k6,k9 andk16 an attacker can use the following equations:

δ₃ =S⁻¹(x9⊕k₉)⊕S⁻¹(x^′₉⊕k₉) 3δ₃ =S⁻¹(x6⊕k₆)⊕S⁻¹(x^′₆⊕k₆) 2δ3 =S⁻¹(x3⊕k3)⊕S⁻¹(x^′₃⊕k3)

δ3 =S⁻¹(x16⊕k16)⊕S⁻¹(x^′₁₆⊕k16)

Finally, in order to obtain information onk4,k7,k10 andk13 an attacker can use the following equations:

δ₄ =S⁻¹(x13⊕k₁₃)⊕S⁻¹(x^′₁₃⊕k₁₃) δ4 =S⁻¹(x10⊕k10)⊕S⁻¹(x^′₁₀⊕k10) 3δ4 =S⁻¹(x7⊕k7)⊕S⁻¹(x^′₇⊕k7) 2δ4 =S⁻¹(x4⊕k4)⊕S⁻¹(x^′₄⊕k4)

It can be noted that the equations have an identical structure, and, therefore, the solutions are of similar nature. An evaluation of each set of equations will be expected to return 2⁸ unique hypotheses for the key bytes concerned. Therefore, an attacker would expect to have 2³²key hypotheses for the secret key used.

3.2 Analysis of the first step of the fault attack

The first step of the fault attack uses four sets of equations to reduce the key space of AES. In this section we determine the expected number of key hypotheses that an attacker will have at each stage of an attack.

In order to analyze the number of valid hypotheses in the first stage of the attack we consider the first set of equations given in Section 3.1. In this set of equations δ₁ is∈ {1, . . . ,255}. Ifδ1 is equal to zero then one could say that the expected fault has not been injected. If δ₁ is zero it would imply thatx₁ is equal tox^′₁ and all 256 key hypotheses are possible. Let us first consider the first equation in this set:

2δ1=S⁻¹(x1⊕k1)⊕S⁻¹(x^′1⊕k1)

We know the values ofx₁ andx^′₁ from the correct and faulty ciphertexts respectively.

For a given value of 2δ1there will 0, 2 or 4 valid key hypotheses. The mean hypotheses for allδ₁∈ {1, . . . ,255}is approximately one, and, therefore, 256 key hypotheses when allδ1 ∈ {1, . . . ,255}are considered.

The same can be said for each of the four equations in the set given above. How- ever, for a given value ofδ1 each of the four equations would be expected to return approximately one hypothesis for a key byte. These values will give one hypothesis for the quartet of key bytes{k1, k₈, k₁₁, k₁₄}. Given that an attacker will have to take into account all the values in {0, . . . ,255} there will be 256 possible values for the quartet{k1, k₈, k₁₁, k₁₄}. After an attacker has analyzed the four equations defined in Section 3.1 there would be an expected 2³²key hypotheses.

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3.3 The Second Step of the Fault Attack

In order to further reduce the key hypotheses we use the relationship between the ninth round key and the tenth round key.

We consider the key-scheduling algorithm (see Algorithm 2), the ninth round key, K9, generates the tenth round key,K10. The key schedule is invertible andK9can be expressed in terms of elements ofK10. The value ofK9 can be expressed as







k1⊕S(k14⊕k10)⊕h10k5⊕k1 k9⊕k5 k13⊕k9

k₂⊕S(k15⊕k₁₁) k₆⊕k₂ k₁₀⊕k₆k₁₄⊕k₁₀ k3⊕S(k16⊕k12) k7⊕k3 k11⊕k7k15⊕k11

k₄⊕S(k13⊕k₉) k₈⊕k₄ k₁₂⊕k₈k₁₆⊕k₁₂





 .

We can observe that the fault values in the first column of the state matrix at the output of the eighth roundMixColumnis (2f^′, f^′, f^′,3f^′), wheref^′ is a non-zero arbitrary value in F2⁸. Using the InverseMixColumn operation and using the inter- relations between the fault values, we can define the following equation:

2f^′=S⁻¹

(

¹⁴

⁽

^S⁻¹^(x¹^⊕^k¹⁾^⊕^((k¹^⊕^S(k¹⁴^⊕^k¹⁰⁾^⊕^h¹⁰⁾⁾

⁾

^⊕¹¹

⁽

^S⁻¹^(x⁸^⊕^k⁸^)⊕

(k2⊕S(k15⊕k11))

)

⊕13

(

^S⁻¹(x11⊕k11)⊕(k3⊕S(k16⊕k12))

)

⊕ 9

(

^S⁻¹(x8⊕k8)⊕(k4⊕S(k13⊕k9))

) )

^⊕^S⁻¹

(

¹⁴

⁽

^S⁻¹^(x^′¹^⊕^k¹⁾

⊕((k1⊕S(k8⊕k10)⊕h10))

)

^⊕¹¹

(

^S⁻¹^(x^′8⊕k8)⊕(k2⊕S(k15⊕k11)

)

^⊕

13

(

^S⁻¹(x^′11⊕k₁₁)⊕(k3⊕S(k16⊕k₁₂))

)

⊕9

(

^S⁻¹(x^′8⊕k₈)⊕

(k4⊕S(k13⊕k₉))

) )

Similarly, we can define the following equations:

f^′=S⁻¹

(

⁹

⁽

^S⁻¹^(x¹³^⊕^k¹³⁾^⊕^(k¹³^⊕^k⁹⁾

⁾

^⊕¹⁴

⁽

^S⁻¹^(x¹⁰^⊕^k¹⁰⁾^⊕^(k¹⁰^⊕^k¹⁴⁾⁾

⁾

^⊕

11

(

^S⁻¹^(x7⊕k7) ⊕(k15⊕k11)

)

^⊕¹³

(

^S⁻¹^(x4⊕k4)⊕(k16⊕k12)

) )

^⊕

S⁻¹

(

⁹

⁽

^S⁻¹^(x13^′ ⊕k₁₃)⊕(k13⊕k₉)

)

⊕14

(

^S⁻¹(x^′₁₀⊕k₁₀)⊕(k10⊕k₁₄))

)

⊕ 11

(

^S⁻¹^(x^′7⊕k7) ⊕(k15⊕k11)

)

^⊕¹³

(

^S⁻¹^(x^′4⊕k4)⊕(k16⊕k12)

) )

f^′=S⁻¹

(

¹³

⁽

^S⁻¹^(x⁹^⊕^k⁹⁾^⊕^(k⁹^⊕^k⁵⁾

⁾

^⊕⁹

⁽

^S⁻¹^(x⁶^⊕^k⁶⁾^⊕^(k¹⁰^⊕^k⁶⁾⁾

⁾

^⊕

14

(

^S⁻¹^(x3⊕k3)⊕(k11⊕k7)

)

^⊕¹¹

(

^S⁻¹^(x16⊕k16)⊕(k12⊕k8)

) )

^⊕

S⁻¹

(

¹³

⁽

^S⁻¹^(x⁹^′ ^⊕^k⁹⁾^⊕^(k⁹^⊕^k⁵⁾

⁾

^⊕⁹

⁽

^S⁻¹^(x^′⁶^⊕^k⁶⁾^⊕^(k¹⁰^⊕^k⁶⁾⁾

⁾

^⊕

14

(

^S⁻¹^(x^′3⊕k3) ⊕(k11⊕k7)

)

^⊕¹¹

(

^S⁻¹^(x^′16⊕k16)⊕(k12⊕k8)

) )

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3f^′=S⁻¹

(

¹¹

⁽

^S⁻¹^(x²^⊕^k²⁾^⊕^(k²^⊕^k¹⁾

⁾

^⊕¹³

⁽

^S⁻¹^(x⁵^⊕^k⁵⁾^⊕^(k⁶^⊕^k⁵⁾⁾

⁾

^⊕

9

(

^S⁻¹^(x12⊕k12) ⊕(k10⊕k9)

)

^⊕¹⁴

(

^S⁻¹^(x15⊕k15)⊕(k14⊕k13)

) )

^⊕

S⁻¹

(

¹¹

⁽

^S⁻¹^(x²^′ ^⊕^k²⁾^⊕^(k²^⊕^k¹⁾

⁾

^⊕¹³

⁽

^S⁻¹^(x^′⁵^⊕^k⁵⁾^⊕^(k⁶^⊕^k⁵⁾⁾

⁾

^⊕

9

(

^S⁻¹^(x^′12⊕k12) ⊕(k10⊕k9)

)

^⊕¹⁴

(

^S⁻¹^(x^′15⊕k15)⊕(k14⊕k13)

) )

The second stage of the attack is coupled with the first stage, and can be used to further reduce the number of key hypotheses.

3.4 Analysis of the second step of the fault attack

The expected number of hypotheses produced by the second step of the attack follows a similar reasoning to the analysis of the first step, given in Section 3.2.

If we consider the second equation defined in Section 3.3, it can be rewritten as f^′=A⊕B ,

whereAandB are defined as

A=S⁻¹

(

⁹

(

^S⁻¹^(x13⊕k13)⊕(k13⊕k9)

)

^⊕

14

(

^S⁻¹^(x10⊕k10)⊕(k10⊕k14))

)

^⊕¹¹

(

^S⁻¹^(x7⊕k7)⊕ (k15⊕k11)

)

^⊕¹³

(

^S⁻¹^(x4⊕k4)⊕(k16⊕k12)

) )

and

B=S⁻¹

(

⁹

(

^S⁻¹^(x^′13⊕k13)⊕(k13⊕k9)

)

^⊕

14

(

^S⁻¹^(x^′10⊕k10)⊕(k10⊕k14))

)

^⊕¹¹

(

^S⁻¹^(x^′7⊕k7)⊕ (k15⊕k11)

)

^⊕¹³

(

^S⁻¹^(x^′4⊕k4)⊕(k16⊕k12)

) )

.

We can considerAandBto be random values inF2⁸. For a given values off^′ the difference between A and B will be equal to f^′ with a probability of ₂¹8. Using the same reasoning, the probability of all four equations being valid is ₂¹8

4

=₂¹32. We have to consider all the possible values of f^′, i.e. {0, . . . ,255}. A given key hypothesis will, therefore, be valid for some arbitrary value off^′with a probability of 2⁸×₂¹32 = ₂¹24.The first step of the attack is expected to return 2³²hypotheses each of which still be under consideration at the end of the second step with a probability of

1

2²⁴.One would, therefore, expect the second step of the attack to produce 2⁸ possible key hypotheses.

3.5 Attacking Other Bytes

In the previous sections we describe an attack where we base our Differential Fault Analysis on the knowledge that a fault has been induced in the first byte of the state matrix. However, we can note that the analysis returns a very small number of hypotheses. We can, therefore, conduct 16 independent analyses under the assumption that a fault is induced each of the 16 bytes of of the state at the beginning of the eighth round. An attacker would expect this to produce 2⁴×2⁸= 2¹²valid key hypotheses, which is still a trivial exhaustive search.

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4 Comparison with Previous Work

There are several versions of fault-based differential cryptanalysis that are able to reduce the number of key hypotheses from two faults injected into an implementation of AES, as described in [5, 9, 12]. However, the analysis proposed in this paper is more effective, since the resulting exhaustive search can be reduced to a trivial size using one fault. The number of key hypotheses returned by previous work would be somewhat time consuming. The advantage of the proposed attack is that it does not need to reproduce a successful attack in order to able to determine a secret key. Acquiring multiple faulty ciphertexts can be problematic as faults are only successful with a certain probability, and the effect cannot always be predetermined. This would mean that an attacker could potentially have to search among numerous faulty ciphertexts to find a pair that both have the desired fault.

5 Conclusion

This paper proposes a fault-based differential cryptanalysis of AES, that is an extended version of the attack described in [9]. An attacker would expect to be able to reduce the number of key hypotheses from 2¹²⁸ to 2⁸ with one well placed fault. As noted in [8], these attacks can be conducted without any knowledge of the plaintext being enciphered, as an attacker would just need to know the plaintexts were the same.

There are many descriptions of a fault-based differential cryptanalysis of AES that could be prevented by repeating the last two or three rounds of an implementation of AES, to verify that no exploitable fault has been inserted [1–3, 12, 13]. However, to prevent the attack described in this paper the last four rounds would need to be repeated to check no fault was injected. Moreover, given how much information can be gleaned from one fault, one would expect there are attacks that require more faulty ciphertexts that would be able to make use of faults in earlier rounds. One would, therefore, suggest that in order to protect an implementation of AES the last five rounds should be protected against fault injection.

Acknowledgements

The work described in this paper has been supported in part by the European Commis- sion IST Programme under Contract ICT-2007-216676 ECRYPT II and EPSRC grant EP/F039638/1 “Investigation of Power Analysis Attacks”. The second author would like to acknowledge the support of Department of Science and Technology (DST) India under the Fast Track Proposals for Young Scientists for the proposal entitled ”Design and Analysis of Side Channel Attack Resistant Symmetric Key Cryptosystems”.‘

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